Solve {l}{y+z=5}{2x+4z=4}{2x-3y=-14}

$$\left. \begin{array} { l } { y + z = 5 } \\ { 2 x + 4 z = 4 } \\ { 2 x - 3 y = - 14 } \end{array} \right.$$

Solve for y, z, x

$$x=-4$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve y+z=5 for y.

$$y=-z+5$$

Substitute -z+5 for y in the equation 2x-3y=-14.

$$2x-3\left(-z+5\right)=-14$$

Solve the second equation for z and the third equation for x.

$$z=1-\frac{1}{2}x x=-\frac{3}{2}z+\frac{1}{2}$$

Substitute 1-\frac{1}{2}x for z in the equation x=-\frac{3}{2}z+\frac{1}{2}.

$$x=-\frac{3}{2}\left(1-\frac{1}{2}x\right)+\frac{1}{2}$$

Solve x=-\frac{3}{2}\left(1-\frac{1}{2}x\right)+\frac{1}{2} for x.

$$x=-4$$

Substitute -4 for x in the equation z=1-\frac{1}{2}x.

$$z=1-\frac{1}{2}\left(-4\right)$$

Calculate z from z=1-\frac{1}{2}\left(-4\right).

$$z=3$$

Substitute 3 for z in the equation y=-z+5.

$$y=-3+5$$

Calculate y from y=-3+5.

$$y=2$$

The system is now solved.

$$y=2 z=3 x=-4$$